Recent works have introduced new equivariant neural networks, motivated by their improved generalization compared to traditional deep neural networks. While experiments support this advantage, the theoretical understanding of their generalization properties remains limited. In this paper, we analyze the generalization capabilities of Group Convolutional Neural Networks (GCNNs) with the ReLU activation function through the lens of Vapnik-Chervonenkis (VC) dimension theory. We investigate how architectural factors—such as the number of layers, weights, and input dimensions—affect the VC dimension. A key challenge in our analysis is proving a lower bound on the VC dimension, for which we introduce new techniques, establishing a novel connection between GCNNs and standard deep neural networks. Additionally, we compare our derived bounds to those known for fully connected neural networks. Our results extend previous findings on the VC dimension of continuous GCNNs with two layers, offering new insights into their generalization behavior, particularly their dependence on input resolution.

We study the generalization capacity of group convolutional neural networks. We identify precise estimates for the VC dimensions of simple sets of group convolutional neural networks. In particular, we find that for infinite groups and appropriately chosen convolutional kernels, already two-parameter families of convolutional neural networks have an infinite VC dimension, despite being invariant to the action of an infinite group.

Symmetries built into a neural network have appeared to be very beneficial for a wide range of tasks as it saves the data to learn them. We depart from the position that when symmetries are not built into a model a priori, it is advantageous for robust networks to learn symmetries directly from the data to fit a task function. In this paper, we present a method to extract symmetries learned by a neural network and to evaluate the degree to which a network is invariant to them. With our method, we are able to explicitly retrieve learned invariances in a form of the generators of corresponding Lie-groups without prior knowledge of symmetries in the data. We use the proposed method to study how symmetrical properties depend on a neural network's parameterization and configuration. We found that the ability of a network to learn symmetries generalizes over a range of architectures. However, the quality of learned symmetries depends on the depth and the number of parameters.

The new user coldstart problem arises when a recommender system does not yet have any information about a user. A common solution to it is to generate a profile by asking the user to rate a number of items. Which items are selected determines the quality of the recommendations made, and thus has been studied extensively. We propose a new elicitation method to generate a static preference questionnaire (SPQ) that poses relative preference questions to the user. Using a latent factor model, we show that SPQ improves personalized recommendations by choosing a minimal and diverse set of questions. We are the first to rigorously prove which optimization task should be solved to select each question in static questionnaires. Our theoretical results are confirmed by extensive experimentation. We test the performance of SPQ on two real-world datasets, under two experimental conditions: simulated, when users behave according to a latent factor model (LFM), and real, in which only real user judgments are revealed as the system asks questions. We show that SPQ reduces the necessary length of a questionnaire by up to a factor of three compared to state-of-the-art preference elicitation methods. Moreover, solving the right optimization task, SPQ also performs better than baselines with dynamically generated questions.